Understanding Point Types in Real Analysis

[STEMucator]

Just a few definitions I would like to verify so I'm not studying the wrong stuff.

Interior Point : A point [itex]Q \in S \subseteq ℝ^n[/itex] is an interior point of S if [itex]\forall \delta > 0, \exists N_{\delta}(Q) \subseteq S[/itex]. The interior of S consists of all interior points and is denoted [itex]S˚[/itex]

Boundary Point : A point [itex]Q \in S \subseteq ℝ^n[/itex] is a boundary point of S if [itex]\forall \delta > 0, \exists P_1 \in S \wedge P_2 \in (ℝ^n - S) \space| \space P_1, P_2 \in N_{\delta}(Q)[/itex]

Limit Point : A point [itex]Q \in S \subseteq ℝ^n[/itex] is a limit point of S if [itex]\forall \delta > 0, \exists P \in S \space | \space P \in N_{\delta}(Q), \space P≠Q[/itex]

Trying to condense my stuff, hopefully I'm doing this correctly.

 

[micromass]

Just a few definitions I would like to verify so I'm not studying the wrong stuff.

Interior Point : A point [itex]Q \in S \subseteq ℝ^n[/itex] is an interior point of S if [itex]\forall \delta > 0, \exists N_{\delta}(Q) \subseteq S[/itex]. The interior of S consists of all interior points and is denoted [itex]S˚[/itex]

It should be [itex]\exists \delta >0[/itex].

Boundary Point : A point [itex]Q \in S \subseteq ℝ^n[/itex] is a boundary point of S if [itex]\forall \delta > 0, \exists P_1 \in S \wedge P_2 \in (ℝ^n - S) \space| \space P_1, P_2 \in N_{\delta}(Q)[/itex]

Limit Point : A point [itex]Q \in S \subseteq ℝ^n[/itex] is a limit point of S if [itex]\forall \delta > 0, \exists P \in S \space | \space P \in N_{\delta}(Q), \space P≠Q[/itex]

OK. But these definitions also hold if you don't demand [itex]Q\in S[/itex].

 

[STEMucator]

It should be [itex]\exists \delta >0[/itex].



OK. But these definitions also hold if you don't demand [itex]Q\in S[/itex].

I thought that it was for all deltas I choose? What if I chose delta so large that my neighborhood was contained in the compliment?

Also, for the purposes of a calc II course, we're assuming Q is inside of the set, so I should be good there for now.